Linear Chebyshev approximation of complex-valued functions

Abstract

Abstract.This paper is concerned with Chebyshev approximation by linear functions to complex-valued data. The problem is nonlinear, and we present a convergent algorithm for its solution. We also pose a related linear problem which is simple to solve, and which produces approximations which are near-best in the Chebyshev sense within a factor of \J2. Some numerical examples are provided.

Wave polynomials, transmutations and Cauchy’s problem for the Klein–Gordon equation

“…It is worth mentioning that despite the kernel K(x, t; h) is not the classical solution of the problem (32)- (33) in the case when q ∈ C [−b, b], nevertheless the function H(u, v) is a classical solution of the problem (36)-(37), see [21]. Define G := ∂H ∂u .…”

“…In the existing algorithms the discretized problem is considered and solved directly as a nonlinear optimization problem, e.g., a convex programming problem [33,34], or the problem is transformed into a semi-infinite programming problem with the use of the fact that |h| = max φ∈[0,2π) Re  e iφ · h  . The dual problem is considered and discretized for the second time with respect to the angle φ and solved by the simplex method [33,35] or by a Remez-like algorithm [36][37][38][39].…”

A new method for Chebyshev approximation of complex-valued functions

“…[2], [12], [13]. Considering the computation of best rational Chebyshev approximations to complex-valued functions by descent algorithms, Ellacott and Williams [6] report that the main portion of computer time is spent in the solution of the linear subproblems.…”

“…This is a problem of nonlinear optimization, and it has been shown by Krabs and Opfer [12], Barrodale, Delves, and Mason [2] how to solve (2) by means of suitably chosen nonlinear programming codes (if 7 is a finite set).…”

“…[2]). Here it is required to approximate f(z) = 1 /(z -2) on the disc |z| < 1 by a polynomial of degree n -1 with real coefficients.…”